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# lecture13 - Physics 2102 Aurora Borealis Jonathan Dowling...

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Unformatted text preview: Physics 2102 Aurora Borealis Jonathan Dowling Physics 2102 L ectu re 1 5 M a g n e t ic f ie ld s Star Quake on a “I’ll be back…. Magnetar! Use of Magnetic Fields in Your Everyday Life! Magnetic Fields Electric fields are created: • microscopically, by electric charges (fields) of elementary particles (electrons, protons) • macroscopically,by adding the field of many elementary charges of the same sign Magnetic fields are created : • microscopically, by magnetic “moments” of elementary particles (electrons, protons, neutrons) • macroscopically, by • adding many microscopic magnetic moments (magnetic materials); or by • electric charges that move (electric currents) Magnetic vs. Electric F o rces We know that an electric fields exists because it accelerates electric charges, with a force independent of the velocity of the charge, proportional to the electric charge: FE = qE We know that a magnetic field exists because it accelerates electric charges in a direction perpendicular to the velocity of the charge, with a magnitude proportional to the velocity of the charge and to the magnitude of the charge: FB= q v x B Magnetic forces are perpendicular to both the velocity of charges and to the magnetic field (electric forces are parallel to the field). Since magnetic forces are perpendicular to the velocity, they do no work! (W=F · r) Speed of particles moving in a magnetic field remains constant in magnitude, the direction changes. Kinetic energy is constant! (no work). C ir c u la r M o tio n : v F r Since magnetic force is transverse to motion, the natural movement of charges is circular. v2 F = ma = m for circular motion r mv 2 Therefore q v B = r mv r= qB B into blackboard. I n g e n e r a l, p a th is a h e lix ( c o m p o n e n t o f v p a r a lle l to f ie ld is u n ch an g ed ). Example Two charged ions A and B traveling with a constant velocity v enter a box in which there is a uniform magnetic field directed out of the page. The subsequent paths are as shown. What can you conclude? Av B v (a) Both ions are negatively charged. (b) Ion A has a larger mass than B. (c) Ion A has a larger charge than B. mv r= qB (d) None of the above. S a m e s p e e d a n d B f o r b o th m a s s e s . S o : io n w ith la r g e r m a s s :c h a r g e r a tio ( m /q ) m o v e s in c ir c le o f la r g e r r a d iu s . B u t th a t ’ s a ll w e k n o w ! Cathode Ray Tube (Old TVs & Computer Monitors) Hot cathode emits electrons Get accelerated by positive plate Might be deflected using plates Produce point of light on screen. In a magnetic field: B rr v!B v Fe Dot shifts sideways. Examples of Motion in Magnetic Fields Aurora borealis (northern lights) Synchrotron Suppose you wish to accelerate charged particles as fast as you can. Linear accelerator (long). Fermilab, Batavia, IL (1km) M a g n e t ic f o r c e o n a w ir e . L L q = it = i vd r rr F = q vd ! B r r iL r r r F = q !B =iL!B q r rr F =iL!B Note: If wire is not straight, compute force on differential elements and integrate: r rr dF = i dL ! B E x a m p le Wire with current i. Magnetic field out of page. What is net force on wire? F1 = F3 = iLB dF = iBdL = iBRd! By symmetry, F2 will only have a vertical component, " " 0 0 F2 = ! sin(# )dF =iBR ! sin(# )d# = 2iBR Ftotal = F1 + F2 + F3 = iLB + 2iRB + iLB = 2iB ( L + R ) Notice that the force is the same as that for a straight wire, and this would be true no matter what the shape of L R R L the central segment!. Example 4: The Rail Gun • Conducting projectile of length 2cm, mass 10g carries constant current 100A between two rails. • Magnetic field B = 100T points outward. • Assuming the projectile starts from rest at t = 0, what is its speed after a time t = 1s? rails B L I projectile • Force on projectile: F= ILB (from F = iL x B) • Acceleration: a = iLB/m (from F = ma) • v(t) = iLBt/m (from v = v0 + at) = (100A)(0.02m)(100T)(1s)/(0.01kg) = 2000m/s = 4,473mph = MACH 8! Rail guns in the “Eraser” movie " R a il g u n s a r e h y p e r - v e lo c ity w e a p o n s th a t s h o o t a lu m in u m o r c la y r o u n d s a t ju s t b e lo w th e s p e e d o f lig h t. I n o u r f ilm , w e 'v e ta k e n e x is tin g s te a lth te c h n o lo g y o n e s te p f u r th e r a n d g iv e n th e m a n X - r a y s c o p e s ig h tin g s y s te m ," n o te s d ir e c to r R u s s e ll. " T h e s e g u n s r e p r e s e n t a w h o le n e w te c h n o lo g y in w e a p o n r y th a t is s till in its in f a n c y , th o u g h a la r g e - s c a le v e r s io n e x is ts in lim ite d n u m b e r s o n b a ttle s h ip s a n d ta n k s . T h e y h a v e in c r e d ib le r a n g e . T h e y c a n p ie r c e th r e e - f o o t th ic k c e m e n t w a lls a n d th e n k n o c k a c a n a r y o f f a tin c a n w ith a b s o lu te a c c u r a c y . I n o u r f ilm , o n e c o n tr a c to r h a s f in a lly d e v e lo p e d a n a s s a u lt- s iz e d r a il g u n . W e r e s e a r c h e d th is q u ite a b it, a n d th e te c h n o lo g y is r e a lly ju s t a r o u n d th e c o r n e r , w h ic h is o n e o f th e e x c itin g p a r ts o f th e s to r y ." W a r n e r B r o s ., p r o d u c tio n n o te s , 1 9 9 6 . http://movies.warnerbros.com/eraser/cmp/prodnotes.html#tech Also: INSULTINGLY STUPID MOVIE PHYSICS: http://www.intuitor.com/moviephysics/ Principle behind electric motors. T o rq u e o n a C u rren t L o o p : Rectangular coil: A=ab, current = i Net force on current loop = 0 But: Net torque is NOT zero! F1 = F3 = iaB F" = F1 sin(! ) Torque = " = F#b = iabB sin(! ) For a coil with N turns, τ = N I A B sinθ, where A is the area of coil Magnetic Dipole Moment We just showed: τ = NiABsinθ N = number of turns in coil A=area of coil. Define: magnetic dipole moment µ r ˆ µ = ( NiA)n r ˆ µ, n Right hand rule: curl fingers in direction of current; thumb points along µ rrr " = µ!B As in the case of electric dipoles, magnetic dipoles tend to align with the magnetic field. Electric vs. Magnetic Dipoles r ˆ µ = ( NiA)n +Q p=Qa -Q QE θ QE rrr " = p! E rrr " = µ!B ...
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